Application of neural networks and neuro-fuzzy models in construction scheduling

Construction scheduling is a complex process that involves a large number of variables, making it difficult to develop accurate and efficient schedules. Traditional scheduling techniques rely on manual analysis and intuition, which are prone to errors and often fail to account for all the variables involved. This results in project delays, cost overruns, and poor project performance. Artificial intelligence models have shown promise in improving construction scheduling accuracy by incorporating historical data, site-specific conditions, and other variables that traditional scheduling methods may not consider. In this research study, application of soft-computing techniques to evaluate construction schedule and control of project activities in order to achieve optimal performance in execution of building projects were carried out. Artificial neural network and neuro-fuzzy models were developed using data extracted from a residential two-storey reinforced concrete framed-structure construction schedule and project execution documents. The evaluation of project performance indicators in earned value analysis from 0 to 100% progress at 5% increment with a total of seventeen tasks were carried out using Microsoft Project software and data obtained from the computation were utilized for model development. Using input–output and curve-fitting (nftool) function in MATLAB, a 6-10-1 two-layer feed-forward network with tansig activation-function (AF) for the hidden neurons and linear AF output neurons was generated with Levenberg–Marquardt (Trainlm) training algorithm. Similarly, with the aid of ANFIS toolbox in MATLAB software, the training, testing and validation of the ANFIS model were carried out using hybrid optimization learning algorithm at 100 epochs and the Gaussian-membership-function (gaussmf). Loss-function parameters namely MAE, RMSE and R-values were taken as the performance evaluation criteria of the developed models. The generated statistical results indicates no significant difference between model-results and experimental values with MAE, RMSE, R2 of 1.9815, 2.256 and 99.9% respectively for ANFIS-model and MAE, RMSE, R2 of 2.146, 2.4095 and 99.998% respectively for the ANN-model. The model performance indicated that the ANFIS-model outclassed the ANN-model with their results satisfactory to deal with complex relationships between the model variables to produce accurate target response. The findings from this research study will improve the accuracy of construction scheduling, resulting in improved project performance and reduced costs.

www.nature.com/scientificreports/ The use of AI in the construction industry benefits both shareholders and investors in all phases of the construction process, including the proposal, costing and financing; material acquisition and correct execution; setup and resource management; and commercial prototype rehabilitation. In order to reduce the demand for experts in structure development and schedule designs, researchers and participants in construction-related projects develop technologies that resemble AI. To complete a project on time and under budget, a great project schedule is essential 29,30 . According to Schelle 31 , effective structure management entails the competent arrangement of several instances of contributing stakeholders, societies and fundamental building blocks. This might involve simulated elements such as tasks, errands and charges, as well as infinite associated units of diverging interactions. For building projects, when given step-by-step instructions and mandatory reinvigoration of responsibilities, this may allow complex undertakings to be managed successfully so that the intended results are achieved. Such recommendations and principles are given for large datasets that have existed over time and are perhaps active. They flow from one correctly specified form to the next properly outlined one. AlTabtabai 32 , for instance, employed a networked BP to launch a managerial method employing specialists chosen from the activities timetable, who supervised and predicted the repurposing of an abandoned many-story building.
This study uses the building of a residential, two-story, reinforced concrete framed structure in Nigeria as a case study to explore the application of artificial intelligence to construction scheduling in order to improve the project duration prediction and achieve cost minimization. Plus, we create an earned-value-management (EVM) model for better forecasting of the progress and performance and to enhance the efficiency of rescheduling and reallocation processes with the use of a decision support system by applying artificial neural networks and adaptive Neuro-fuzzy inference. A contractor's bid and a construction timetable are equivalent. The timetable represents the estimated time necessary to complete the project, much like the bid represents the estimation of the cost that is assumed to be required to complete the project 33 . By using the building of a residential, two-story, reinforced concrete framed structure as a case study, this study aims to illustrate how artificial intelligence can be applied to construction scheduling in order to improve the project duration prediction and for cost minimization in Nigeria's construction industry.
Additionally, earned-value-management (EVM) model is developed for better forecasting of the progress and performance and to enhance the efficiency of rescheduling and reallocation processes with the decision support system. A bid from contractors and a construction timetable are equivalent. Similar to how the bid is the estimate of the costs necessary to accomplish the project, the schedule denotes the anticipated amount of time needed to complete the project 34 . By using this, other stakeholders and general contractors can keep track of a project's overall progress.
This study is aimed at applying artificial intelligence to construction scheduling to achieve better prediction of the project duration and minimize the costs in the building construction industry. The details derived from this research study will provide a new dynamic monitoring and optimization tool to track the progress of a project. The purpose of the research is to investigate the potential of neural networks and Neuro-fuzzy models in improving construction scheduling accuracy and efficiency and to provide insights into the application of these models in the broader field of construction engineering and management. A good construction project schedule is accurate, thorough and updated frequently, with communication regarding the project given first importance. Team cooperation is another important element since it helps tasks to be completed successfully. Scheduling allows project managers to match the labor, supplies, equipment and all other resources connected with activities and construction tasks over time, which is essential for the completion and success of a construction project. A well-planned construction schedule ensures the completion of projects by outlining the exact pace at which each job is to be completed, the sequences and methods for delivering resources, and the execution of all generated tasks 35,36 . Significance of the study. The application of neural networks and neuro-fuzzy models in construction scheduling is significant for several reasons. First, construction projects are complex and involve multiple tasks that need to be completed in a specific sequence. Any delay in one task can have a cascading effect on the rest of the project. Therefore, accurate scheduling is critical for the success of a construction project. Secondly, traditional scheduling methods rely on the experience and intuition of project managers, which can be subjective and lead to errors. The use of artificial intelligence (AI) models, such as neural networks and neuro-fuzzy models, can provide objective and data-driven scheduling solutions. Thirdly, the construction industry has been slow to adopt new technologies, and the application of AI in construction scheduling represents a step forward in the adoption of digital technologies. The use of AI models can help improve productivity, reduce project delays, and ultimately save costs. Overall, the significance of this study lies in its contribution to the development of more accurate and efficient scheduling methods for the construction industry, which can lead to improved project outcomes and better resource utilization.
Project scheduling process. The timetable for a construction project provides a clear view of all the project milestones, due dates and timelines. It should be regularly updated to measure the progress and show the various steps that must be taken before completion. The contractor's ideas on how to complete the project are fully explained and demonstrated in a construction project schedule, which also clarifies the scope of the job. The necessary duration and the work activities are represented sequentially in the work scope. A project schedule is the only management document that can predict when a project will be completed, which is an important fact to be aware of. Scheduling involves the description of specific tasks and activities, as well as accomplishments that show a start date and an expected end date 37 . It is impossible to overstate how important scheduling is to a project's success in construction. An effective timetable may be able to guarantee that the project is finished on time and within budget. It involves how and when a task is completed as well as how quickly the work is done. www.nature.com/scientificreports/ Furthermore, scheduling specifies the process and technique for material delivery. Finally, it allows for seasonal readjustments so that changes and uncertainties can be taken into account 28 . The task setup and timetabling rudiments can be divided into eight practicable stages, which enables timely execution within the designed budget as illustrated in Fig. 1.

Methodology
The research study was carried out to examine feedback from the building/construction industry on the applications, utilization and feasibility of artificial intelligence in construction project scheduling using the established tender document. A deductive methodology was employed because it was best suited to the problem characteristics, and a qualitative approach was taken due to the investigative nature of the study. The study commenced with us conducting an in-depth literature review of relevant and recent scheduling methodologies used in the construction industry, with an emphasis on identifying their benefits and limitations 38 . The broad categories considered were design, procurement and execution. The information derived from the tender document serves as the foundation for periodic work valuation, variation valuation, variance reconciliation and all cost-related activities during building construction 39 . A project can be divided into several stages, with each representing a group of activities that culminate in the completion of one or more of the achievements, after usually having been completed in the order listed. This structuring divides the project into reasonable subdivisions for stressfree managing, designing and control. Depending on the nature of the project, each has different stages. The number of stages or the need for them is determined by several factors, including the project's size, complexity and potential impact 40,41 . Importantly, a tool for project management planning and analysis of the schedule is required given the relationships and interdependencies between the project's activities. The critical path method, which is used in this research methodology, can be used to plan a large-scale activity network for project progress and management 42 . The start and end times of activities in the original schedule plan may be impacted, and the critical path may barely be reflected. Overcoming such issues, the critical path method is a project scheduling and analysis method that represents the tasks that must be completed in a specific project, including the trade-off between activity duration and cost. The basic rule is that any increase in critical activity duration leads to an increase in critical activity cost. The research methodology flowchart is shown in Fig. 2 43,44 . Steps to the critical path calculation. The critical path method (CPM) determines the task's shortest achievable completion time using the project actions' potential start and end times. In fact, more managers now view the critical path scheduling strategy as the most useful and practical scheduling technique. The duration denotes the shortest amount of time required to complete a certain project. If there is a barrier on the vital path, more time will be required before the project is completed. In order to use the critical path scheduling method in practice, construction task planners must act as a resource constraint via a precedence relationship 44,45 . The steps for calculating the CPM are stated below: Forward-scrolling algorithm. This presents calculations for the critical path starting from the beginning of the node to the end of the grid, using Eq. (1) where D ij is the activity duration, E(i) is the earliest start time for a given activity and E(j) is the latest start time 46 . www.nature.com/scientificreports/ Backward-scrolling algorithm. This is the opposite of the front-scrolling algorithm. It calculates from the last node of the activity network and returns to the foremost node using mathematical relationships, as presented in Eq.
where L(j) is the latest end time for a given activity and L(i) is the earliest end time for a given activity. The difference (time) between the early start and the late start is known as elasticity, which represents the time in which the activity can be delayed without affecting the required project duration 40,48 . Using a two-story residential building project as a case study, the precedence, relations and durations for 17 activities required for the project are presented in Figs. 3 and 4.

Calculating earned value. Earned value management (EVM) depicts in straightforward words the level
of coverage and what tasks remain in a project. This accurate report is critical in recognizing faults, changing plans, amending mistakes and ensuring not only timely but also excellent delivery. The EVM puts cost and time on a unified scale, allowing one to graphically evaluate the actual work done vs. what was expected. The following direct indicators are adopted to appropriately scrutinize the timetable and costs accrued for a given mission using EVM 49 .
• Planned value (PV): is otherwise called the budgeted cost of work scheduled (BCWS). It is the cost sum through the current reporting period. It is the projected rate of a task arranged to conclude within an agreed interval 50 ; • Actual cost (AC): is also called the actual cost of work performance (ACWP). The actual cost implies the authentic payments made to complete a task by the set date. It is the recorded cost of completed works when using the preset interval alone; • Earned value (EV): is otherwise referred to as the budgeted cost of work performance (BCWP). This is the aggregate task financial plan, increased by the percentage of task achievement. It denotes the accepted financial plan of tasks completed by the deadline 51 ; • Schedule Performance Index (SPI) and Schedule Variance (SV): the SPI is the ratio of EV to PV. It is a comparative quota of the project's interval adeptness, which compares the actual headway to the premeditated headway. An SPI rate of < 1.0 designates that less work has been completed than anticipated, while a value of > 1.0 designates that more tasks were completed than were scheduled 35 . The SV is the variance flanked by www.nature.com/scientificreports/ the authentic tasks delivered contrary to the guesswork. It tells us whether the project is within plans or not. Zero variance depicts a project running according to the timetable/schedule, while a negative or positive difference depicts arrears or getting ahead of schedule, respectively 35 . The mathematical relationships are presented in Eqs. (3) and (4):  www.nature.com/scientificreports/ • Cost Performance Index (CPI) and Cost Variance (CV): the CPI is the ratio of EV to AC. It is a comparative quota of the cost of the project in terms of proficiency, which is capable of guesstimating the price of tasks left uncompleted 52 . The CV, therefore, stands for the variance between EV and AC. Whether a project is carried out as budgeted is showcased by the EV and AC. Zero indicates that the project is falling within the appropriated cost margins, whereas the project is considered as over or under the appropriate cost if the difference is negative or positive. The mathematical relationships are presented in Eqs. (5) and (6)  Model performance evaluation. The performance of the intelligent model developed was evaluated in order to confirm that it has a proven ability to predict or estimate the target parameters with an acceptable degree of accuracy. Several performance criteria (statistical measures) used in the related literature, such as the loss function parameters, mean absolute error (MAE) and root mean square error (RMSE), are given with the formulas shown in Eqs. (7) and (8) [55][56][57] .
where n is the size of the data points under investigation, E i is the actual or experimental results and M i is the estimated model values.

Results, discussion and analysis
The schedule computation was carried out using Microsoft Project and Microsoft Excel software in line with the research carried out by Dayal 58 for effective management of varying sizes of construction projects. The construction project under study was executed by a medium-sized firm with a planned duration of 95 days at an estimated direct cost of 25.8 million naira. The description of the project consisted of a residential, two-story, reinforced concrete framed structure with five bedrooms and a penthouse. The general information on the project was reviewed and the reasons for delaying the completion of the work. The critical and flexible activities involved in the project are presented in Table 1 from the computed results. The flow of the construction work's 17 activities and dependencies indicated little or insignificant difference between the earliest and latest finish points of the project activities in the initial stages. However, as the project proceeded to the advanced stages, the relationships between the events and activities signaled appreciable slack periods, which provided necessary time for the safe completion of clashing preceding activities in the project 58,59 . The performance indicators' computation results were extracted and are presented in Table 2, showing the actual time (AT), schedule variance (SV), earned value (EV), actual cost (AC), schedule performance indicator   www.nature.com/scientificreports/ Pearson's correlation. According to previous studies, Pearson's correlation coefficients, as presented in Table 3, were deployed to evaluate the linear relationship between the predictors and explanatory variables. The results indicated strong positive relationships between the target response factor, earned value (EV), and the following performance indicators: planned progress, actual time (AT), earned schedule (ES), actual cost (AC) and cost variance (CV). Meanwhile, negative linear relationships were observed to exist for the schedule performance indicator and schedule variance factors 64,65 .

Artificial neural network (ANN) model development. The modeling process was carried out with
the datasets fed to the neural network using MATLAB software. The model framework was designed as six input variables namely, ES, planned progress, SV, SPI, CPI and AT; with one output parameter as the EV. The processing parameter settings for the neural network model are presented in Table 4 and Fig. 7, which show a 6-10-1 two-layer feed-forward network with a tansig activation function (AF) for the hidden neurons and linear AF output neurons. This can perform multidimensional mapping to solve complex system solutions. In order to determine the best-performing n-neurons, mean squared error (MSE) and R-values, evaluation criteria were used, which revealed that 10 neurons produced optimal results 66,67 .
Training state of the ANN. The ANN training state plot (plottrainstate) of the neural network indicated a gradient of 26.6334, with the optimal value computed at 15 epochs. The validation checks failed at six because the errors were repeated six times before the process finally stopped. This represented the best performance of the neural network; at that stage, its performance ceased to improve further. The error function was repeated at zero points from epochs 0-9, then rose linearly from one to six over epochs 10-15. However, starting from epoch 10, we observed overfitting of the data. Therefore, epoch 9 was taken as the baseline, and its weight functions were selected as the final weights, as shown in Fig. 8 68 .
Validation performance of the ANN. The mean square error (MSE) was the criteria tool used to evaluate the model's performance while randomly selecting different hidden neuron numbers, activation function parameters and training algorithms for validation of the ANN network, as shown in Fig. 9. The graphical results indicated the best validation performance of 4.3639 at epoch 9 for the optimized network (8-10-1). The results indi- www.nature.com/scientificreports/ cated a satisfactory performance of the ANN model. It was capable of predicting the target response parameters accurately by generalizing the sets of complex input variables with minimum error 69,70 .
Error histogram of the ANN. An error histogram for the simulated smart model performance is presented in Fig. 10, which illustrates the level of correlation between the experimental and predicted variables with a 20 bins error histogram for training, testing and validation of the network. The zero-error point indicates the best performance during the simulation. Almost 95% of the data yield an error of less than 1%. The zero error is indicated with a yellow line in the middle at 0.04565 for the error function, with 50, 55 and 65 instances in the training, validation and testing sets, respectively 21,71 .
Regression plot of the ANN. A regression plot presents the model relationships for the actual data and the ANN model results using the coefficient of determination and mean squared error (MSE) for the training, validation and testing sets, as shown in Fig. 11. The smart model output results were plotted on the y-axis of the regression plot while the actual values were on the x-axis. The derived statistical results show a satisfactory performance in terms of the prediction accuracy of the ANN model with 0.9996, 0.9945 and 0.92232 results obtained for training, testing and validation, respectively 72 . Table 5. The criteria used for performance evaluation of the network were the mean squared error (MSE), root mean squared error (RMSE) and coefficient of determination (r 2 ). The

Neuro-fuzzy model development. Detailed computation results showing the performance indicators of
the cost and schedule for the project were utilized to build the ANFIS model input-output constraints appropriately. The earned schedule (ES), planned progress (percent), schedule variance (SV), schedule performance index (SPI), cost performance index (CPI) and actual time (AT) in weeks were the six input variables, and the earned value was the output variable (EV). The model variables' relationships, showing the input-output associations, are given in Fig. 12 74 . The ANFIS model was trained, tested and validated using the ANFIS toolbox in MATLAB software. The MATLAB software workspace and membership function were generated using the sub-clustering fuzzy-inference-system formulation method after system datasets were loaded into it. Moreover, a hybrid method of optimization was deployed as the learning algorithm, which was adopted to train the fuzzy inference at 100 epochs 75 . Table 6 shows the learning and membership function constraints for data treatment, with an error tolerance value of 0, range of impact value of 0.5 and squash factor, reject and accept ratios of 1.25, 0.15 and 0.5, respectively. The Gaussian membership function (gaussmf) was utilized to evaluate the degrees of belongingness of the factors, as presented in Eq. (10). The model variables can be represented as follows:  Testing and training ANFIS. To achieve the training, validation and testing of the neuro-fuzzy network using the prescribed hybrid optimization training methods and FIS constraints, the datasets used for the neuro-fuzzy  www.nature.com/scientificreports/  www.nature.com/scientificreports/ modeling procedure were separated and arranged in two parts. The datasets were loaded from the workspace for ANFIS network training with one output and four input variables, as well as the graphical plot of 20 indices for network training. Training and testing error results of 8.0523 and 6.4218, respectively, were calculated in the process, as presented in Figs. 13 and 14 76 .
Graphical plots of the membership function. Graphical plots that show the membership function for the model variables in the ANFIS network were generated by means of the MATLAB recreation toolbox, which was used to robotically advance the suitable connection function standards to increase the records' generality. Figure 15 shows the membership function designs, with the variety of records for model constraints on the x-axis and the discourse value from 0 to 1 on the y-axis 77 .

Selection of the optimized ANFIS model. A comparison table illustrating varying ANFIS network architectures
and their respective performance using RMSE, MSE, and r 2 is shown in Table 7. The optimized ANFIS model after network training and testing was the architecture type with a Gaussian membership function. The training performance results for the optimized model were 8.0523, 2.84 and 0.99999 for the MSE, RMSE and r 2 , respectively. Plus, for the testing performance, the optimized model produced values of 6.4218, 2.534 and 0.99999 for the MSE, RMSE and r 2 , respectively.

ANFIS model variables' graphical expression.
A soft computing smart model was installed for the evaluation of the schedule performance indicators. This studies the generality of statistic sets it has been served with assistance from a hybrid optimization set of rules. Such a model has the power to precisely pair a given collection of inputs with the matching yield value. With a three-area apparent design, the prototypical variables' interactions are weighed to spot their substantial one-to-one significance or possession, as revealed in Fig. 16. The influence of the independent variables on the earned value is assessed in this process 78 .

Model validation.
The developed smart intelligent model's prediction performance was evaluated using a statistical method and loss function parameters, namely the mean absolute error (MAE) and root mean square   Table 8. The loss function statistical computation, which offered a good evaluation criterion for the performance of the developed smart intelligent model, is shown in

Sensitivity analysis. Sensitivity analysis assesses the contribution of the individual independent variables
to the output response (EV). For this purpose, the methods reported by Razavi et al. 80 were adapted to determine which inputs had the greatest impact on the output variable. We used the relevancy factor (r), where r is in the range of [− 1, 1]. The r values were calculated using Eq. (11).
where X k,i and Y i are the ith input and output, respectively; Y and X k are the average values of the output and kth input, respectively; and n denotes the total number of data points. The computation results are presented in Figs

Conclusions
The research assessment of the application of artificial intelligence in construction scheduling for efficient project management was achieved in this study with a two-story residential structure construction taken as a case study to design and evaluate the schedule and cost performance indicators. The following conclusions can be drawn: • The construction project under study was executed by medium-sized firm with a planned duration of 95 days at an estimated direct cost of 25.8 million naira. The project performance indicators were evaluated through earned value analysis from 0-100% progress, at 5% increments, with a total of 17 tasks. This was carried out using Microsoft Project software, and data obtained from the computation were utilized for model development; • Pearson's correlation results obtained for the model variables indicated strong positive relationships between the response factor, earned value (EV), and the following performance indicators: planned progress, actual time (AT), earned schedule (ES), actual cost (AC) and cost variance (CV). Meanwhile, negative linear relationships were observed to exist for the schedule performance indicator and schedule variance factors; • Data generated in this process were expertly selected for the input-output model variables' formulation to improve project performance, reduce costs, and enhance overall project management. ANN and ANFIS were deployed for the smart modeling process using MATLAB software for the model simulation, training, testing and validation; • The model's prediction accuracy was evaluated using loss function parameters, namely the root mean squared error (RMSE) and mean absolute error (MAE  www.nature.com/scientificreports/ adaptive and robust, dealing with complex relationships between the model variables to produce an accurate target response; • The results suggest that these models can be effectively integrated into existing scheduling processes and have the potential to significantly improve project performance. The developed models also offer a viable and accurate means of providing project performance indicators that enable project/construction managers to proficiently monitor, control and execute projects with the designed quality, time and resources. Furthermore, details derived through this research study will contribute toward developing an essential template for efficient planning and accountability of construction projects, to prevent challenges such as cost overruns. www.nature.com/scientificreports/   www.nature.com/scientificreports/ Study's limitations and recommendations. Investigative research on construction schedule evaluation using artificial intelligence tools is very important given that it can be applied to deal with non-linear complex problems better than conventional statistical approaches. The gains derived from this work will contribute essential information to the decision-making process in construction planning, monitoring and controlling, to achieve the optimum solution. The system datasets utilized for the smart intelligent modeling in this research study were, however, limited to two-story residential structure construction in the area of the study. Therefore, further investigation is recommended using different classes of buildings based on the intended use of the structure, along with the deployment of multiple hybrid AI algorithms such as the neural networks-genetic-fuzzylogic hybrid algorithm.

Data availability
The datasets generated and/or analyzed during the current study are available from the corresponding author on reasonable request. www.nature.com/scientificreports/